\Input $A$ (sparse matrix), $b$ (right-hand side)
\Output $x$ (solution vector)\vspace{0.2cm}
\State Set the initial guess $x^0$
- \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{kryl}$)} \label{algo:conv}
+ \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsarm}$)} \label{algo:conv}
\State $x^k=Solve(A,b,x^{k-1},max\_iter_{kryl})$ \label{algo:solve}
\State retrieve error
\State $S_{k~mod~s}=x^k$ \label{algo:store}
- \If {$k$ mod $s=0$ {\bf and} error$>\epsilon_{kryl}$}
+ \If {$k$ mod $s=0$ {\bf and} error$>\epsilon_{tsarm}$}
\State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
\State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:}
\State $x^k=S\alpha$ \Comment{compute new solution}
equals to the restart number of the GMRES-like method. Moreover, a tolerance
threshold must be specified for the solver. In practice, this threshold must be
much smaller than the convergence threshold of the TSARM algorithm (i.e.
-$\epsilon$). Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the
+$\epsilon_{tsarm}$). Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the
solution $x_k$ into the column $k~ mod~ s$ of the matrix $S$. After the
minimization, the matrix $S$ is reused with the new values of the residuals. To
solve the minimization problem, an iterative method is used. Two parameters are
To summarize, the important parameters of TSARM are:
\begin{itemize}
-\item $\epsilon_{kryl}$ the threshold to stop the method of the krylov method
+\item $\epsilon_{tsarm}$ the threshold to stop the TSARM method
\item $max\_iter_{kryl}$ the maximum number of iterations for the krylov method
\item $s$ the number of outer iterations before applying the minimization step
\item $max\_iter_{ls}$ the maximum number of iterations for the iterative least-square method
The following parameters have been chosen for our experiments. As by default
the restart of GMRES is performed every 30 iterations, we have chosen to stop
-the GMRES every 30 iterations (line \ref{algo:solve} in
-Algorithm~\ref{algo:01}). $s$ is set to 8. CGLS is chosen to minimize the
-least-squares problem. Two conditions are used to stop CGLS, either the
-precision is under $1e-40$ or the number of iterations is greater to $20$. The
-external precision is set to $1e-10$ (line \ref{algo:conv} in
-Algorithm~\ref{algo:01}). Those experiments have been performed on a Intel(R)
-Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
+the GMRES every 30 iterations, $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is
+chosen to minimize the least-squares problem with the following parameters:
+$\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$. The external precision is set to
+$1e-10$ (i.e. ). Those experiments
+have been performed on a Intel(R) Core(TM) i7-3630QM CPU @ 2.40GHz with the
+version 3.5.1 of PETSc.
In Table~\ref{tab:02}, some experiments comparing the solving of the linear
-In the following we describe the applications of PETSc we have experimented. Those applications are available in the ksp part which is suited for scalable linear equations solvers:
+In the following we describe the applications of PETSc we have
+experimented. Those applications are available in the ksp part which is suited
+for scalable linear equations solvers:
\begin{itemize}
-\item ex15 is an example which solves in parallel an operator using a finite difference scheme. The diagonal is equals to 4 and 4
- extra-diagonals representing the neighbors in each directions is equal to
- -1. This example is used in many physical phenomena , for exemple, heat and
- fluid flow, wave propagation...
-\item ex54 is another example based on 2D problem discretized with quadrilateral finite elements. For this example, the user can define the scaling of material coefficient in embedded circle, it is called $\alpha$.
+\item ex15 is an example which solves in parallel an operator using a finite
+ difference scheme. The diagonal is equals to 4 and 4 extra-diagonals
+ representing the neighbors in each directions is equal to -1. This example is
+ used in many physical phenomena , for exemple, heat and fluid flow, wave
+ propagation...
+\item ex54 is another example based on 2D problem discretized with quadrilateral
+ finite elements. For this example, the user can define the scaling of material
+ coefficient in embedded circle, it is called $\alpha$.
\end{itemize}
For more technical details on these applications, interested reader are invited
to read the codes available in the PETSc sources. Those problem have been
